As with non-deterministic machines, this can also be viewed as a deterministic machine with an extra input, which corresponds to the random selections.

There are several different ways of defining what it means for a random Turing machine to accept or reject an input. Let ProbT⁡(x) be the probability that T halts in an accepting state when the input is x.

A positiveone-sided error machine is said to accept x if ProbT⁡(x)≥12 and to reject x if ProbT⁡(x)=0. A negative one-sided error machine accepts x if ProbT⁡(x)=1 and rejects x if ProbT⁡(x)≤12. So a single run of a positive one-sided error machine never misleadingly accepts but may misleadingly reject, while a single run of a negative one-sided error machine never misleadingly rejects.

The definition of a positive one-sided error machine is stricter than the definition of a non-deterministic machine, since a non-deterministic machine rejects when there is no certificate and accepts when there is at least one, while a positive one-sided error machine requires that half of all possible guess inputs be certificates.

A two-sided error machine accepts x if ProbT⁡(x)≥23 and rejects x if ProbT⁡(x)≤13.

A minimal error machine accepts x if ProbT⁡(x)>12 and rejects x if ProbT⁡(x)<12.

One additional variant defines, in addition to accepting states, rejecting states. Such a machine is called zero error if on at least half of all inputs it halts in either an accepting or a rejecting state. It accepts x if there is any sequence of guesses which causes it to end in an accepting state, and rejects if there is any sequence of guesses which causes it to end in a rejecting state. In other words, such a machine is never wrong when it provides an answer, but does not produce a decisive answer on all input. The machine can emulate a positive (resp. negative) one-sided error machine by accepting (resp. rejecting) when the result is indecisive.

It is a testament to the robustness of the definition of the Turing machine (and the Church-Turing thesis) that each of these definitions computes the same functions as a standard Turing machine. The point of defining all these types of machines is that some are more efficient than others, and therefore they define different time and space complexity classes.